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Convergence to infinitely divisible distributions with finite variance for someweakly dependent sequences

Published online by Cambridge University Press:  15 November 2005

Jérôme Dedecker  and
Sana Louhichi
Jérôme Dedecker
Affiliation:
Laboratoire de Statistique Théorique et Appliquée, Université Paris 6, Site Chevaleret, 13 rue Clisson, 75013 Paris, France; dedecker@ccr.jussieu.fr
Sana Louhichi
Affiliation:
Laboratoire de Probabilités, Statistique et modélisation, Université de Paris-Sud, Bât. 425, 91405 Orsay Cedex, France; Sana.Louhichi@math.u-psud.fr
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Abstract

We continue the investigation started in a previous paper, onweak convergence to infinitely divisible distributions with finitevariance. In the present paper, we study this problem for someweakly dependent random variables, including in particularassociated sequences. We obtain minimal conditions expressed interms of individual random variables. As in the i.i.d. case, wedescribe the convergence to the Gaussian and the purelynon-Gaussian parts of the infinitely divisible limit. We alsodiscuss the rate of Poisson convergence and emphasize the specialcase of Bernoulli random variables. The proofs aremainly based on Lindeberg's method.

Keywords

Infinitelydivisible distributionsLévy processesweak dependenceassociationbinary random variablesnumber of exceedances.

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Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 9 , February 2005 , pp. 38 - 73
Copyright
© EDP Sciences, SMAI, 2005

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